# Simple Events are Mutually Exclusive

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## Theorem

Let $\EE$ be an experiment.

Let $e_1$ and $e_2$ be distinct simple events in $\EE$.

Then $e_1$ and $e_2$ are mutually exclusive.

## Proof

By definition of simple event:

\(\ds e_1\) | \(=\) | \(\ds \set {s_1}\) | ||||||||||||

\(\ds e_2\) | \(=\) | \(\ds \set {s_2}\) |

for some elementary events $s_1$ and $s_2$ of $\EE$ such that $s_1 \ne s_2$.

It follows that:

\(\ds e_1 \cap e_2\) | \(=\) | \(\ds \set {s_1} \cap \set {s_2}\) | Definition of $e_1$ and $e_2$ | |||||||||||

\(\ds \) | \(=\) | \(\ds \O\) | Definition of Set Intersection |

The result follows by definition of mutually exclusive events.

$\blacksquare$

## Sources

- 1965: A.M. Arthurs:
*Probability Theory*... (previous) ... (next): Chapter $2$: Probability and Discrete Sample Spaces: $2.2$ Sample spaces and events